Mathetics the third
If at first you don't succeed, try again...twice
Twice I have tried to explain Tom Gilbert’s system of curriculum design, mathetics. Each attempt has been a failure.
The first attempt was based on applications of mathetics conducted by the Draper Correctional Center in Elmore, Alabama, in the 1960s. In what I believe to be a wonderful representation of a boom in instructional design through the 50s and 60s, a branch of the Alabama correctional system invested time and money into designing training materials for convicts. They had an honest-to-god “Materials Development Unit,” making illustrated guides to learning vocational skills. This was a brief postwar era when advances in instructional design had convinced people that well-designed curriculum could solve the problems of education.
Ultimately I think I mischaracterized aspects of mathetics. Worse than that, I think it was a boring read.
I started over again, and this time I made sure to use the very small amount of Tom Gilbert’s work that I could find on the topic. Still, it seemed like I couldn’t find enough of Gilbert’s own work. Didn’t he author any programs? What about this seldom-mentioned Journal of Mathetics? Frankly, mathetics is (in part) a system of design – but it doesn’t seem like the sources I found were able to use that system to explain…mathetics. What is it even supposed to be?
And then, in plain sight, I found the only 2 issues ever published of The Journal of Mathetics.
There’s a great story that BF Skinner tells about his rare foray into applied work. Upon visiting his daughter Deborah’s fourth grade class in 1953, he was big mad that the teaching sucked. The teacher, like the pigeon, was not to blame; rather he saw that the teacher as the only teaching stimulus couldn’t work for an entire classroom. According to Julie Vargas, it was later that same day that he invented a version of his teaching machine – a simple device that could give immediate feedback when a student selected a multiple-choice option. Sidney Pressey had developed such a machine in the mid-1920s, though it appears to never have been particularly popular, and Skinner believed his own machine was superior.
Somewhere in this timeframe, Skinner was creating Programmed Instruction, a style of writing that carefully presented text that required readers to provide an answer, and then check their answer against the author’s answer, before proceeding. His linear format typically utilized simple sentences with fill-in-the-blanks style. An example, from the first page of Holland & Skinner (1961):
Technically speaking, a reflex involves an eliciting stimulus in a process called elicitation. A stimulus ______ a response. [on the next page is printed elicits]
The units were very small (typically 1-3 sentences), and the material was carefully analyzed so that the reader built on acquired knowledge as they proceeded. The material was supposed to be field-tested, to ensure that readers could generally get the correct answer. (Here is an online version of Holland & Skinner designed by BDS Solutions).
As early as 1958 (the book also has a web version), 3 years before linear Programmed Instruction, Norman Crowder happened to develop and popularize the most impactful form of Programmed Instruction: the branching style (or as he often called it, the intrinsic style). Here he is describing his method (and criticizing “teaching machines”) in 1960, the year before Skinner’s linear style was published. Of the two styles, branching has decisively captured the market. Why did Crowder beat Skinner in the realm of public opinion? Simple:
Shortly after the explosion of educational Programmed Instruction in the 60s (approximately 1976), there was a long-running and immensely popular series of “choose your own adventure” books using the same branching format. Less famous is Crowder’s series of books, TutorText, which taught readers various subjects using the branching format, ceasing publication around 1972 after 32 volumes.
Crowder wasn’t a scientist per se – how did he develop a system to rival that of Skinner? It all began with World War 2.
World War 2 was a big-budget sequel to World War 1, and as such America drafted 8 million men in 5 years to the Army alone. From the previous link, an initial 13 weeks of training was eventually expanded to 17 weeks, covering nearly 600 hours of instruction. In a certain light, this was the largest public education experiment ever conducted by the American government.
Ultimately, who were the people leading this education? If we take the example of Bob Mager, it was not “educators” – according to Bob, he was promoted to trainer due to his superior marksmanship. The military was quickly assembled with men who skewed younger, who at times were there against their will, who quite literally would have represented an “average” aptitude, and who needed to be taught to perform in matters of life and death. A captive audience of people who don’t want to learn – a perfect and perfectly horrible experiment.
The feedback for teachers like Mager was immediate. If a group of 18-year-old men couldn’t shoot a target, you knew. If they didn’t understand your written or spoken instructions, you’d hear it from them. If they died in battle, you’d hear that too. And so Mager quickly refined his teaching methods. The focus was on performing a task, which had to be narrowly and clearly defined – “knowing” was simply not as important as “doing.” Writing had to be simple and clear, preferably illustrated.
Mager would later adopt the branching style in his books on Programmed Instruction. Perhaps it’s no surprise that Crowder was a product of the Air Force, where he would differentiate his classroom instruction based on errors made by learners. This was wholly applied work, and the remediation of errors was not based on data or a specific educational theory; Crowder would just pick a different path for a learner after an error. Still, the idea itself is brilliant in its simplicity: if you are teaching a skill to a student who makes an error, you may give them some type of corrective feedback. Inserting corrective feedback into written work was a side effect of pressing non-teachers into teaching roles.
The 1960s were when veterans of WW2 (and Skinner) turned their expertise in training towards the education market. Tom Gilbert, due to a childhood health condition, missed the draft. But his path crossed with Skinner and Susan Markle, setting the occasion to develop his own system of design in the early 1960s, when market interest was at an all-time high. By 1962, he published the first (and penultimate) issue of The Journal of Mathetics, detailing his system that he had reportedly used for some time prior.
Strictly speaking, mathetics was not a system for written material alone, and in that way differed from Programmed Instruction. Rather, it was intended to be a comprehensive training system that could teach virtually any skill.
The skill itself was carefully analyzed, in particular the outcome or product;
the steps were precisely outlined based on the behaviors of the learner;
and the skill was taught with backwards chaining.
Sounds easy enough! Well, it was actually a little more complicated.
The work should be field-tested and adjusted (common in Programmed Instruction and in Direct Instruction – it led to superior products but inflated costs)
The work should take into account the population to be trained
There may be illustrations (at the time, required hiring an illustrator)
There may be practice with real or dummy items, such as machinery
Also I wasn’t totally honest when I said that the skill is “carefully analyzed.” Gilbert outlines a process so thorough, yet so general, that I feel dizzy reading it. There’s a considerable amount of Mechner notation. Maybe I’m not smart enough to understand the system, but based on the lack of general adoption, at least I’m in the majority.
The clearest evidence that mathetics was used in the 1960s is the work of the aforementioned Draper Correctional Center in Alabama. The leaders of that project have positive things to say about mathetics, and share practical considerations for producing material. Even these superfans had this to say about learning mathetics:
Did mathetics fail? Certainly I would guess that Tom Gilbert imagined a bigger impact. And it never exerted an influence in the public imagination like branching Programmed Instruction. Even within the canon of Tom Gilbert’s work, Human Competence probably has had the greater impact.
Obviously one point of failure is the lack of a clear, comprehensive, readily available instructional material on mathetics. While Gilbert himself appears to have published a mercifully short volume in 1969, it might have been a bit late to market. (I also haven’t found a copy anywhere).
Perhaps adding to the challenges is the fact that Gilbert himself was a bit of a prickly guy. Undoubtedly brilliant, I’m not sure he sold his system brilliantly.
Lastly, and perhaps most importantly, is my own pet theory on Programmed Instruction, Direct Instruction, and other quality work: cost. These systems stressed competent, thorough writing and planning, as well as field-testing and revision. The field-testing meant that writers could catch problems with their design; imagine you write a question that 95% of your test group fails – the question may have a fault. However, these excellent programs can be cheaply mimicked and put into the market without field tests. The result is a similar-looking product that costs less. To the untrained eye, there’s no discernible difference.
If mathetics didn’t make an oversized impact, we sometimes hear its echoes. Janet Twyman, known for her impact on behavior analysis within education, recommends mathetics alongside Engelmann as late as 2021. Authors discussing instructional design mention it in 2024 (even if, in this case, the author appears to disagree philosophically).
Perhaps making a big impact with huge nerds is the best possible outcome. From the vantage point of 2024, it seems that the 50s and 60s contained enormous optimism for instructional design. In the ensuing decades, that optimism faded – both for logical reasons (excellent instructional materials aren’t perfect) and for illogical reasons (philosophical disagreements; see the previous paragraph). As a person living today I can see that a popular movement towards Direct Instruction, or Programmed Instruction, or mathetics, was never going to happen. But does it matter if average moms and dads are teaching their toddlers in mathetical terms? Or would we rather it make an impact on instructional designers, who may be at the forefront of programming for the myriad online programs that currently serve up a supremely mixed bag of results?
Here’s hoping that the next generation of nerds finds value in the lessons of the past.
Excellent article. I enjoyed reading it.